On the limit distribution of Frobenius numbers

Abstract

The Frobenius number g(a) of an integer vector a with positive coprime coefficients is defined as the largest integer that does not have a representation as a non-negative integer linear combination of the coefficients of a. According to a recent result by Marklof, if a is taken to be random in an expanding d-dimensional domain D, then (a1 · · · a d ) −1/(d−1) g(a) has a limit distribution. In the present paper we prove an asymptotic formula for the (algebraic) tail behavior of this limit distribution. We also prove that the corresponding upper bound on the probability of the Frobenius number being large holds uniformly with respect to the expansion factor of the domain D. Finally we prove that for large d, the limit distribution of (a1 · · · a d ) −1/(d−1) g(a) has almost all of its mass concentrated between (d − 1)! 1/(d−1) and 1.757 · (d − 1)! 1/(d−1) . The techniques involved in the proofs come from the geometry of numbers, and in particular we use results by Schmidt on the distribution of sublattices of Z m , and bounds by Rogers and Schmidt on lattice coverings of space with convex bodies.

Integer matrices, sublattices of $$\mathbb {Z}^{m}$$ Z m , and Frobenius numbers

“…Now when C ≤ t n / (Log t) n , then C t n /(Log C) n , hence t −n (Log C) 2n−2 C −1 (Log C) n−2 , and the first summand in (20.6) is (up to a factor 1) bounded by the estimate in (20.2). Altogether, we get As had been noted in Corollary 1 of [24], N (t, T ) = 0 unless t < mT 1−1/n , i.e. T 1/n > (t/m) 1/(n−1) .…”

“…Then Aliev et al [4] showed this to be true with (t) = t −2 m−1 m+1 +ε (see the footnote in [24], p. 85).…”

“…Therefore Remark Strömbergsson in the proof of Theorem 3 of [24] uses similar arguments when dealing with λ 1 , . .…”

Integrality Gaps of Integer Knapsack Problems

“…Proof. The inequality (5.3) immediately follows from Theorem 3 in [26] applied with D = [0, 1] n−1 .…”

“…How large is the integer programming gap of a "typical" knapsack problem? To tackle this question we will utilize the recent strong results of Strömbergsson [26] (see also Schmidt [24] and references therein) on the asymptotic distribution of Frobenius numbers. The main result of this paper will show that for any ǫ > 2/n the ratio Gap c (A) A ǫ ∞ c 1 is bounded, on average, by a constant that depends only on dimension n. Hence, for fixed n > 2 and a "typical" integer knapsack problem with large A ∞ , its linear programming relaxation provides a drastically better approximation to the solution than in the worst case scenario, determined by the optimal upper bound (1.6).…”